7 Answers
7

I'm a native speaker and it's actually not that difficult, even though they are quite similar.

While "die Folge" is more like the consequence (so something that follows which doesn't finish it completely, like a secondary result I guess), the word "das Ergebnis" is more like a (final) outcome which finishes (for example a research) completely.

I'd like to add that Folge has multiple meanings, e.g. episode (of TV series), sequence (math), ... Basically it's something that follows in some way.
As a connotation there may be something that follows in turn, making it not final as explained in the other answers.

Ergebnis is often more specific, e.g. you'd say that things falling towards the center of earth is die Folge of the law of gravity. Whereas the Ergebnis of some calculation is 42. If you want to emphasize that the result is obtained by a logical sequence of thoughts:

Das Ergebnis folgt aus Anwendung von (1) und (2)

in that case, you can also say

Die Folge aus (1) und (2) ist ...

You can describe situations where the result "happens", independent of which of a number of theoretically possible alternatives is actually right:

Good catch. "Folge" indeed seems to be mainly used for negative consequences (so does the English word, too?). It's worth noting, however, that the verb "folgen" and the adverbs "folglich" and "infolgedessen" are not restricted to negative statements.
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Em1Jul 24 '14 at 7:04

I think "Ergebnis" is used mainly for intended consequences/results, while "Folge" is used more for unintended consequences (which may explain what's observation that it is most often used for negative consequences: Usually negative consequences are unintended, while people like to claim positive consequences of their actions as intended, even if they didn't really foresee them).

An example of a positive use of "Folge" where I would not say "Ergebnis":